Angle at the centre $x = 2a$centre angle $=2\times$ circumference angle (same arc) Angle in a semicircle $90°$the angle in a semicircle (on a diameter) is a right angle Same segment $a = a$angles in the same segment (on the same arc) are equal Cyclic quadrilateral $a + c = 180°$opposite angles of a cyclic quadrilateral add to $180°$ Tangent & radius $90°$a tangent meets a radius at a right angle Two tangents $PA = PB$two tangents from a point are equal (isosceles)
Alternate segment $a = a$tangent-chord angle $=$ angle in the alternate segment Angle at centre · angle at C $=40°$
centre $=2\times$ circumference
x $=2\times 40°=80°$
Cyclic quadrilateral · angle A $=85°$
opposite angles add to $180°$
angle C $=180°-85°=95°$
Semicircle + triangle · diameter, base angle $35°$
angle in semicircle $=90°$
x $=180°-90°-35°=55°$
Chord: a straight line joining two points on the circle.
Tangent: a line that touches the circle at exactly one point.
Segment: the region between a chord and an arc (major or minor).
Cyclic quadrilateral: a four-sided shape with all four vertices on the circle.
Subtend: a chord or arc "opens up" an angle at a point on the circle.
✗ Halving instead of doubling for the centre
✓ the centre angle is the bigger one: centre $=2\times$ circumference.
✗ Cyclic quad: adding adjacent angles to $180°$
✓ it is opposite angles that add to $180°$.
✗ Giving an angle with no reason
✓ state the theorem name every time – no reason, no method mark.
✗ Alternate segment on the wrong angle
✓ the tangent-chord angle equals the angle in the other (alternate) segment.
Examiners’ reports flag• Could not supply the correct circle-theorem justification, missing alternate-segment or tangent-radius reasoning, or quoting a wrong…
• Could not produce a complete geometric proof
• Always give a reason (the theorem name) with each angle – the reason earns the method mark.
• Two radii make an isosceles triangle (radii are equal), so its base angles are equal.
• Hunt for a diameter or a tangent-meets-radius first – both hand you a free $90°$.
• Mark every angle you can find on the diagram as you go.
• Look for a diameter ($\to 90°$) or a radius meeting a tangent ($\to 90°$) first.
• Two radii $\to$ isosceles triangle $\to$ equal base angles.
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